Math 115. Homework #7 Due Monday 18 October

Section 2.8 (Page 107): 16, 19, 20, 21, 22 Additional problem #1. (This problem gives another proof of Lemma 2.34. Your solutions should not rely on this lemma, and ideally should not use ideas from its proof.)

Let m ∈ ℤ_{>0} . Suppose that a and b have orders h and k, respectively, modulo m, and that gcd(h, k) = 1 .

(a). Let r, s ∈ ℕ. Show that if a^{r}b^{s} ≡ 1 (mod m) , then a^{r} ≡ 1 (mod m) and

b^{s} ≡ 1 (mod m) .

(b). Use the result of part (a) to show that ab has order hk. Section 2.9 (Page 114): 1a, 2, 7 Section 3.1 (Page 135): 3, 7, 11